First, not to rain on your parade, but I think it would cost to much and you would not sell enough. BUUUUT, great thinking and don't stop coming up with stuff. That's how cool things get invented.
As for the idea itself, I think a whole removable top would be the way to go. A bat handle would screw on for normal games, than a handle with a button in to would screw on for button games. That way even if you had a cap, it wouldn't work it's way off during gaming.